SSC CGL 2024 Tier-I · 26 September 2024, Shift 2 · Quantitative Aptitude · Question 12 of 100

Text as printed on a candidate's response sheet for this shift; answer per SSC's tentative key as marked on that sheet. Solution last reviewed 10 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. The questions and options here are reproduced as printed on a candidate's own response sheet for this shift — the Commission's text, unchanged. Each answer is SSC's tentative key as marked on that sheet; SSC allowed challenges to it and never publishes a final key. The solutions are written by UnlockIAS.

If 2a − b = 3 and 8a³ − b³ = 999, then find the value of 4a² − b².

(a)61
(b)65
(c)67
(d)63

Answer

Why

Correct — D. The stem is an image: if 2a − b = 3 and 8a³ − b³ = 999, find 4a² − b². Every term is built from 2a and b, since 8a³ = (2a)³ and 4a² = (2a)².

Use x³ − y³ = (x − y)³ + 3xy(x − y) with x = 2a, y = b:
999 = 3³ + 3 · (2a)(b) · 3
999 = 27 + 9 · (2a)(b) → (2a)(b) = 108

(2a + b)² = (2a − b)² + 4(2a)(b) = 9 + 432 = 441 → 2a + b = 21
4a² − b² = (2a − b)(2a + b) = 3 × 21 = 63 → option (d)

Why the others are wrong

(a)
61 — 61 is not a multiple of 3, and 4a² − b² factors as 3(2a + b), which always is. Accepting it would need 2a + b = 61 ⁄ 3.
(b)
65 — 65 fails the same divisibility test. The cube condition pins 2a + b at exactly 21, and 3 × 21 is 63, not 65.
(c)
67 — 67 is also not divisible by 3. Note that (2a + b)² came out as 441 exactly, so 2a + b = 21 with no rounding anywhere to absorb the extra 4.

Concept

Two identities carry the whole question.

x³ − y³ = (x − y)³ + 3xy(x − y) turns a known difference plus a difference of cubes into the product xy.

(x + y)² = (x − y)² + 4xy then converts that product into the sum.

The move most candidates miss is the substitution x = 2a. Once you see 8a³ as (2a)³ and 4a² as (2a)², the question is about the single pair (2a, b) and never about a and b separately.

(2a + b)² = 441 leaves two branches. The positive one, 2a + b = 21, gives 2a = 12 and b = 9, so a = 6 and b = 9 — check 8(216) − 729 = 999.

The negative branch, 2a + b = −21, gives 2a = −9 and b = −12, which also satisfies both conditions but lands on −63. The paper does not offer −63, so the options settle the sign.

Key facts

x³ − y³ = (x − y)³ + 3xy(x − y), so a difference and a difference of cubes together fix the product xy.
(x + y)² = (x − y)² + 4xy converts a known difference and product into the sum.
4a² − b² = (2a − b)(2a + b) = 3(2a + b) here, so the answer has to be a multiple of 3.
One pair satisfying both conditions is a = 6, b = 9, which gives 144 − 81 = 63 directly.

Study next

The identities a³ ± b³ written in terms of (a ± b)³Recovering xy from a difference and a difference of cubesSpotting a hidden substitution such as x = 2a in a cubic condition

Common traps

Treating the question as being about a and b when every term is built from 2a and b.
Using x³ − y³ = (x − y)(x² + xy + y²), reaching x² + xy + y² = 333, and stopping before extracting xy.
Taking the square root of 441 without checking which sign the options allow.

SSC gives one linear condition and one cubic condition and asks for the quadratic in between, so the identity does the work and the variables are never solved for.

A close cousin is at 10 Sep 2024, 09:00, Quant Q.22, where the sum of three numbers and the sum of their squares are given and a³ + b³ + c³ − 3abc is wanted.

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